Smoothed partially linear quantile regression with nonignorable missing response

In this paper, we propose a smoothed estimator and variable selection method for partially linear quantile regression models with nonignorable missing responses. To address the identifiability problem, a parametric propensity model and an instrumental variable are used to construct sufficient instrumental estimating equations. Subsequently, the nonparametric function is approximated by B-spline basis functions and the kernel smoothing idea is used to make estimation statistically and computationally efficient. To accommodate the missing response and apply the popular empirical likelihood (EL) to obtain an unbiased estimator, we construct bias-corrected and smoothed estimating equations based on the inverse probability weighting approach. The asymptotic properties of the maximum EL estimator for the parametric component and the convergence rate of the estimator for the nonparametric function are derived. In addition, the variable selection in the linear component based on the penalized EL is also proposed. The finite-sample performance of the proposed estimators is studied through simulations, and an application to HIV-CD4 data set is also presented.